Upper Bounds On the ML Decoding Error Probability of General ...
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Upper Bounds On the ML Decoding Error Probability of General Codes over AWGN
arXiv:1308.3303v2 [cs.IT] 26 Dec 2013
Channels Qiutao Zhuang, Jia Liu, and Xiao Ma, Member, IEEE
Abstract In this paper, parameterized Gallager’s first bounding technique (GFBT) is presented by introducing nested Gallager regions, to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The three well-known bounds, namely, the sphere bound (SB) of Herzberg and Poltyrev, the tangential bound (TB) of Berlekamp, and the tangential-sphere bound (TSB) of Poltyrev, are generalized to general codes without the properties of geometrical uniformity and equal energy. When applied to the binary linear codes, the three generalized bounds are reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in the literatures.
Index Terms Additive white Gaussian noise (AWGN) channel, Gallager’s first bounding technique (GFBT), general codes, maximum-likelihood (ML) decoding, parameterized GFBT, trellis code.
I. I NTRODUCTION In most scenarios, there do not exist easy ways to compute the exact decoding error probabilities for specific codes and ensembles. Therefore, deriving tight analytical bounds is an important research subject in the field of coding theory and practice. Since the early 1990s, spurred by the successes of the near-capacity-achieving codes, renewed attentions have been paid to the The authors are with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou 510006, China. J. Liu is also with the College of Information Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou 510225, China. (Email: [email protected]) May 11, 2014
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performance analysis of the maximum-likelihood (ML) decoding algorithm. Though the ML decoding algorithm is prohibitively complex for most practical codes, tight bounds can be used to predict their performance without resorting to computer simulations. As mentioned in [1], most bounding techniques have connections to either the 1965 Gallager bound [2–5] or the 1961 Gallager bound [6–18] based on Gallager’s first bounding technique (GFBT). However, most previously reported upper bounds are focusing on binary linear codes. For binary linear codes modulated by binary phase shift keying (BPSK), there are two main properties, which are geometrical uniformity and equal energy. The geometrical uniformity allows us to make an assumption that the all-zero codeword is the transmitted one, while the property of equal energy is critical to derive the tangential bound (TB) [6] and the tangential-sphere bound (TSB) [10]. For general codes without these two properties, performance analysis becomes more difficult than that for binary linear codes. In this paper, we present parameterized GFBT by introducing nested Gallager regions to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The main contributions as well as the structure of this paper are summarized as follows. 1) We present in Sec. II the parameterized GFBT for general codes. We also present a necessary and sufficient condition on the optimal parameter, and a sufficient condition (with a simple geometrical explanation) under which the optimal parameter does not depend on the signalto-noise ratio (SNR). 2) Within the general framework based on the introduced nested Gallager regions, three existing upper bounds, the sphere bound (SB) of Herzberg and Poltyrev [9], the tangential bound (TB) of Berlekamp [6] and the tangential-sphere bound (TSB) of Poltyrev [10], are generalized in Sec. III to general codes without the properties of geometrical uniformity and equal energy. The three upper bounds are then applied to binary linear codes and reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al. [7] [8], which was rarely cited in the literatures. 3) We use in Sec. IV terminated trellis codes [19] to illustrate how to calculate the parameterized Gallager first bounds on the frame-error probability. Numerical results are also presented in Sec. IV. Sec. V concludes this paper.
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II. T HE PARAMETERIZED G ALLAGER ’ S F IRST B OUNDS A. General Codes A general code C(n, M) ⊂ Rn , in this paper, means a set that contains M n-dimensional real vectors (referred to as codewords). The squared Euclidean distance between a codeword s and the origin point O of the n-dimensional space, denoted by ksk2 , is also referred to as the energy of this codeword. If all codewords have the same energy, we say that the code has the property of equal energy. Given a codeword s, we denote Aδd |s the number of codewords having the Euclidean distance δd with s. We define ∆
Aδd =
X s
Pr{s}Aδd |s ,
(1)
which is the average number of ordered pairs of codewords with Euclidean distance δd . Definition 1: The Euclidean distance enumerating function of a general code C(n, M) is defined as ∆
A(X) =
X
2
Aδd X δd ,
(2)
δd
where X is a dummy variable and the summation is over all possible distance δd . For a general code, there exist at most M2 non-zero coefficients {Aδd }, which is referred to as the Euclidean distance spectrum.
To derive tangential bounds, we also need another distance spectrum for general codes. Given a codeword s with energy δd21 , we denote Bδd1 ,δd2 ,δd |s the number of codewords sˆ having energy δd22 and the Euclidean distance δd with s. We define X ∆ Bδd1 ,δd2 ,δd = Pr{s}Bδd1 ,δd2 ,δd |s ,
(3)
s
which is the average number of ordered pairs of codewords with the Euclidean distance δd and energies δd21 and δd22 , respectively. Definition 2: The triangle Euclidean distance enumerating function of a general code C(n, M) is defined as ∆
B(X, Y, Z) =
X
2
2
2
Bδd1 ,δd2 ,δd X δd1 Y δd2 Z δd ,
(4)
δd1 ,δd2 ,δd
where X, Y, Z are three dummy variables. We call {Bδd1 ,δd2 ,δd } the triangle Euclidean distance spectrum of the given code. May 11, 2014
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B. The Conventional Union Bound Suppose that a codeword s = (s0 , s1 , · · · , sn−1 ) ∈ C(n, M) is transmitted over an AWGN channel. Let y = s+z be the received vector, where z is a vector of independent Gaussian random variables with zero mean and variance σ 2 . For AWGN channels, the maximum-likelihood (ML) decoding is equivalent to finding the nearest codeword ˆs to y. The decoding error probability Pr{E} is Pr{E} =
X
Pr{s}Pr{E|s},
(5)
s
where Pr{E|s} is the conditional decoding error probability when transmitting s over the channel. As usual, we assume that each codeword s is transmitted with equal probability, that is Pr{s} = 1/M. With this assumption, the code rate is is
P
s
ksk2
nM σ2
log M n
and the signal-to-noise ratio (SNR)
.
The conventional union bound on the ML decoding error probability of a general code C(n, M) is Pr{E} =
X
Pr{s}Pr{E|s}
s
δd Aδd |s Q Pr{s} ≤ 2σ s δd XX δd Pr{s}Aδd |s Q = 2σ s δd X δd Aδd Q = , 2σ δ X
X
(6)
d
where Q
δd 2σ
is the pair-wise error probability with Z +∞ z2 1 ∆ √ e− 2 dz. Q(x) = 2π x
(7)
The union bound is simple since it involves only the Q-function and does not require the code structure other than the Euclidean distance spectrum. However, the union bound is loose and even diverges in the low-SNR region. One way to solve this issue is to use the GFBT / R|s}, Pr{E|s} ≤ Pr{E, y ∈ R|s} + Pr{y ∈
(8)
where E denotes the conditional error event, y denotes the received signal vector, and R denotes an arbitrary region around the transmitted signal vector s. The first term in the right hand May 11, 2014
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side (RHS) of (8) is usually bounded by the conditional union bound, while the second term in the RHS of (8) represents the probability of the event that the received vector y falls outside the region R, which is considered to be decoded incorrectly even if it may not fall outside the Voronoi region [20] [21] of the transmitted codeword. For convenience, we call (8) R-bound. Intuitively, the more similar the region R is to the Voronoi region of the transmitted signal vector, the tighter the R-bound is. Therefore, both the shape and the size of the region R are critical to GFBT. Given the region’s shape, one can optimize its size to obtain the tightest R-bound. Different from most existing works, where the size of R is optimized by setting to be zero the partial derivative of the bound with respect to a parameter (specifying the size), we will propose an alternative method by introducing nested Gallager’s regions in the subsection II-D. C. Binary Linear Codes For a binary linear block code C(n, M) of dimension k = log2 M, length n, and minimum Hamming distance dmin, suppose that a codeword c is modulated by binary phase shift keying (BPSK), resulting in a bipolar signal vector s with st = 1−2ct for 0 ≤ t ≤ n−1. Without loss of generality, we assume that the code C has at least three non-zero codewords, i.e., its dimension
k > 1, and the transmitted codeword is the all-zero codeword c(0) (with bipolar image s(0) ).
Let cˆ (with bipolar image sˆ) be a codeword of Hamming weight d, then the Euclidean distance √ between s(0) and sˆ is δd = 2 d. We define ∆
Ad = Aδd |s(0) ,
(9)
which is the number of codewords with Hamming weight d. Since the constellation of binary linear block code is geometrically uniform and each codeword is assumed to be transmitted with equal probability, we have Pr{E} =
X
Pr{s}Pr{E|s}
s
= Pr{E|s(0) } √ ! X d , ≤ Ad Q σ d
(10)
where {Ad } is the weight distribution of the code C.
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D. GFBT with Parameters In this subsection, we will present parameterized GFBT by introducing nested Gallager regions with parameters so that Gallager bounds can be extended to general codes conveniently. To this end, let {R(r), r ∈ I ⊆ R} be a family of Gallager’s regions with the same shape and parameterized by r ∈ I. For example, the nested regions can be chosen as a family of ndimensional spheres of radius r ≥ 0 centered at the transmitted codeword s. We make the following assumptions. Assumptions. A1. The regions {R(r), r ∈ I ⊆ R} are nested and their boundaries partition the whole space Rn . That is,
R(r1 ) ⊂ R(r2 ) if r1 < r2 , ∂R(r1 ) and
\
(11)
∂R(r2 ) = ∅ if r1 6= r2 ,
Rn =
[
∂R(r),
(12)
(13)
r∈I
where ∂R(r) denotes the boundary surface of the region R(r). A2. Define a functional R : y 7→ r whenever y ∈ ∂R(r). The randomness of the received vector y then induces a random variable R. We assume that R has a probability density function (pdf) g(r). A3. We also assume that Pr{E|y ∈ ∂R(r), s} can be upper-bounded by a computable upper bound fu (r|s). For ease of notation, we may enlarge the index set I to R by setting g(r) ≡ 0 for r ∈ / I.
Under the above assumptions, we have the following parameterized GFBT 1 . Proposition 1: For any r ∗ ∈ R, Pr{E|s} ≤ 1
Z
r∗
fu (r|s)g(r) dr + −∞
Z
+∞
g(r) dr.
(14)
r∗
Strictly speaking, we need one more assumption that fu (r|s) is measurable with respect to g(r).
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Proof: Pr{E|s} = Pr{E, y ∈ R(r ∗ )|s} + Pr{E, y ∈ / R(r ∗ )|s} / R(r ∗ )|s} ≤ Pr{E, y ∈ R(r ∗ )|s} + Pr{y ∈ Z r∗ Z +∞ fu (r|s)g(r) dr + ≤ g(r) dr. −∞
r∗
An immediate question is how to choose r ∗ to make the above bound as tight as possible? A natural method is to set the derivative of (14) with respect to r ∗ to be zero and then solve the equation. In this paper, we propose an alternative method for gaining insight into the optimal parameter. Before presenting a necessary and sufficient condition on the optimal parameter, we need emphasize that the computable bound fu (r|s) may exceed one. We also assume that fu (r|s) is non-trivial, i.e., there exists some r such that fu (r|s) ≤ 1. For example, fu (r|s) can be taken as the union bound conditional on y ∈ ∂R(r). Theorem 1: Assume that fu (r|s) is a non-decreasing and continuous function of r. Let r1 be a parameter that minimizes the upper bound as shown in (14). Then r1 = sup{r ∈ I} if fu (r|s) < 1 for all r ∈ I; otherwise, r1 can be taken as any solution of fu (r|s) = 1. Furthermore, if fu (r|s) is strictly increasing in an interval [rmin , rmax ] such that fu (rmin |s) < 1 and fu (rmax |s) > 1, there exists a unique r1 ∈ [rmin , rmax ] such that fu (r1 |s) = 1. Proof: The second part is obvious since the function fu (r|s) is strictly increasing and continuous, which is helpful for solving numerically the equation fu (r|s) = 1. To prove the first part, it suffices to prove that neither r0 < sup{r ∈ I} with fu (r0 |s) < 1 nor r2 with fu (r2 |s) > 1 can be optimal. Let r0 < sup{r ∈ I} such that fu (r0 |s) < 1. Since fu (r|s) is continuous and r0 < sup{r ∈ I},
we can find I ∋ r ′ > r0 such that fu (r ′ |s) < 1. Then we have
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Z
r0
fu (r|s)g(r) dr +
−∞
=
Z
r0
fu (r|s)g(r) dr +
−∞
>
Z
r0
fu (r|s)g(r) dr +
−∞
=
Z
r′
fu (r|s)g(r) dr +
−∞
Z Z Z Z
+∞
g(r) dr r0 r′
g(r) dr +
+∞
Z
g(r) dr
r′
r0 r′
fu (r|s)g(r) dr +
+∞
Z
g(r) dr
r′
r0 +∞
g(r) dr, r′
where we have used the fact that fu (r|s) < 1 for r ∈ [r0 , r ′ ]. This shows that r ′ is better than r0 . Suppose that r2 is a parameter such that fu (r2 |s) > 1. Since fu (r|s) is continuous and nontrivial, we can find r1 < r2 such that fu (r1 |s) = 1. Then we have Z r2 Z +∞ fu (r|s)g(r) dr + g(r) dr −∞ r1
= > =
Z
−∞ Z r1
−∞ Z r1
r2 r2
fu (r|s)g(r) dr + fu (r|s)g(r) dr + fu (r|s)g(r) dr +
Z
r1 Z r2
fu (r|s)g(r) dr +
Z
+∞
g(r) dr
r2
g(r) dr +
r1 Z +∞
Z
+∞
g(r) dr
r2
g(r) dr,
r1
−∞
where we have used a condition that fu (r|s) > 1 for r ∈ (r1 , r2 ], which can be fulfilled by choosing r1 to be the maximum solution of fu (r|s) = 1. This shows that r1 is better than r2 . Corollary 1: Let fu (r|s) be a non-decreasing and continuous function of r. If fu (r|s) does not depend on the SNR, then the optimal parameter r1 minimizing the upper bound (14) does not depend on the SNR, either. Proof: It is an immediate result from Theorem 1. Theorem 1 requires fu (r|s) to be a non-decreasing and continuous function of r, which can be fulfilled for several well-known bounds. Without such a condition, we may use the following more general theorem. Theorem 2: For any measurable subset A ⊂ I, we have Z Z Pr{E|s} ≤ fu (r|s)g(r) dr + r∈A
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g(r) dr.
(15)
r ∈A /
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Within this type, the tightest bound is Z Pr{E|s} ≤
fu (r|s)g(r) dr +
r∈I0
Z
g(r) dr,
(16)
r ∈I / 0
where I0 = {r ∈ I|fu (r|s) < 1}. Equivalently, we have Z min{fu (r|s), 1}g(r) dr. Pr{E|s} ≤
(17)
r∈I
Proof: Let G =
S
r∈A
∂R(r), we have
/ G|s} Pr{E|s} ≤ Pr{E, y ∈ G|s} + Pr{y ∈ Z Z = fu (r|s)g(r) dr + g(r) dr. r∈A
r ∈A /
/ Define A0 = {r ∈ A|fu (r|s) < 1} and A1 = {r ∈ A|fu (r|s) ≥ 1}. Similarly, define B0 = {r ∈ A|fu (r|s) < 1} and B1 = {r ∈ / A|fu (r|s) ≥ 1}. Noticing that Z Z Z fu (r|s)g(r) dr + g(r) dr fu (r|s)g(r) dr ≥ r∈A0 r∈A1 r∈A Z Z Z g(r) dr, fu (r|s)g(r) dr + g(r) dr ≥ r ∈A /
r∈B1
r∈B0
we have Z
≥ = =
Z
Z
Z
fu (r|s)g(r) dr + r∈A
Z
r ∈A /
r∈A0
S
fu (r|s)g(r) dr + B0
fu (r|s)g(r) dr + r∈I0
g(r) dr Z r∈A1
Z
S
g(r) dr
B1
g(r) dr
r ∈I / 0
min{fu (r|s), 1}g(r) dr. r∈I
E. Conditional Pair-Wise Error Probabilities Let δd denote the Euclidean distance between s (the transmitted codeword) and a codeword sˆ. The pair-wise error probability conditional on the event {y ∈ ∂R(r)}, denoted by p2 (r, δd ), is
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p2 (r, δd ) = Pr ky − sˆk ≤ ky − sk | y ∈ ∂R(r) R f (y) dy sk≤ky−sk, y∈∂R(r) ky−ˆ R , = f (y) dy y∈∂R(r)
(18) DRAFT
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where f (y) is the pdf of y. Noticing that, different from the unconditional pair-wise error probabilities, p2 (r, δd ) may be zero for some r. We have the following lemma. Lemma 1: Suppose that, conditional on y ∈ ∂R(r), the received vector y is uniformly distributed over ∂R(r). Then the conditional pair-wise error probability p2 (r, δd ) does not depend on the SNR. Proof: Since f (y) is constant for y ∈ ∂R(r), we have, by canceling f (y) from both the numerator and the denominator of (18), p2 (r, δd ) =
R
sk≤ky−sk,y∈∂R(r) ky−ˆ
R
y∈∂R(r)
dy
dy ,
(19)
which shows that the conditional pair-wise error probability can be represented as a ratio of two “surface area” and hence does not depend on the SNR. Theorem 3: Let fu (r|s) be the conditional union bound, that is, X Aδd |s p2 (r, δd ). fu (r|s) =
(20)
δd
Suppose that, conditional on y ∈ ∂R(r), the received vector y is uniformly distributed over ∂R(r). If fu (r|s) is a non-decreasing and continuous function of r, then the optimal parameter r1 minimizing the bound (14) does not depend on SNR but only on the distance spectrum of the code. Proof: From Lemma 1, we know that fu (r|s) does not depend on the SNR. From Corollary 1, we know that r1 does not depend on the SNR. More generally, without the condition that fu (r|s) is a non-decreasing and continuous function of r, the optimal interval I0 defined in Theorem 2 does not depend on the SNR, either.
F. General Framework of Parameterized GFBT From the above subsection, we can see that there are three main steps to derive a parameterized GFBT. First, choose properly nested regions specified by a parameter. Second, find the pdf of the parameter. Finally, find a computable upper bound on the conditional decoding error probability given that the received vector falls on the boundary of a parameter-specified region. The key of the third step is to find the “projection” of the codewords to the boundary. Here, the “projection” May 11, 2014
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sˆ
r
d
2
s
Fig. 1. The geometric interpretation of the SB for general codes.
means that the intersection between the perpendicular bisector of the segment s sˆ (s and sˆ are the transmitted codeword and decoded codeword, respectively) and the boundary. III. S INGLE -PARAMETERIZED U PPER B OUNDS
FOR
G ENERAL C ODES
For a general code, the property of geometrical uniformity may not hold. As a result, we can not assume a particular transmitted codeword and must average over all conditional error probabilities. In this section, we will first derive the conditional upper bound of Pr{E|s} when transmitting the codeword s over the channel according to the framework of the parameterized GFBT, and then obtain the upper bound of Pr{E} from (5). A. The Parameterized Sphere Bound 1) Nested Regions: The parameterized SB chooses the nested regions to be a family of ndimensional spheres centered at the transmitted signal vector s, that is, R(r) = {y | ky−sk ≤ r}, where r ≥ 0 is the parameter. See Fig. 1 for reference. 2) Probability Density Function of the Parameter: The pdf of the parameter is r2
2r n−1 e− 2σ2 g(r) = n n n , r ≥ 0. 2 2 σ Γ( 2 )
(21)
3) Conditional Upper Bound: The parameterized SB chooses fu (r|s) to be the conditional union bound when transmitting the codeword s over the channel. Given that ||y − s|| = r, y is uniformly distributed over ∂R(r). Hence the conditional pair-wise error probability p2 (r, δd ) May 11, 2014
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does not depend on the SNR and can be evaluated as the ratio of the surface area of a spherical cap to that of the whole sphere. That is, n R arccos( δd ) n−2 ) √ Γ( 2n−1 2r sin φ dφ, r > π Γ( 2 ) 0 p2 (r, δd )= 0, r ≤
δd 2 δd 2
.
Then the conditional union bound is given by X fu (r|s) = Aδd |s p2 (r, δd ).
(22)
(23)
δd
4) The Parameterized SB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (r|s), 1}g(r) dr.
(24)
0
From (1), we define fu (r) ,
X
Pr{s}fu (r|s)
s
=
X
Pr{s}
s
=
δd
=
δd
XX X
X
s
Aδd |s p2 (r, δd )
Pr{s}Aδd |s p2 (r, δd )
Aδd p2 (r, δd ).
(25)
δd
From (5), the parameterized SB for general codes can be written as X Pr{s}Pr{E|s} Pr{E} = s
≤
X
≤
Z
=
Z
Pr{s}
+∞
min +∞ 0
+∞
min {fu (r|s), 1} g(r) dr
0
s
0
Z
(
X
)
Pr{s}fu (r|s), 1 g(r) dr
s
min {fu (r), 1} g(r) dr,
(26)
which is determined by the Euclidean distance spectrum {Aδd }.
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5) Reduction to Binary Linear Codes: For binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between a codeword sˆ √ with Hamming weight d and s(0) is δd = 2 d. Therefore, from (9), (22) and (23), the conditional union bound fu (r|s(0) ) can be written as fu (r|s(0) ) =
X δd
=
Aδd |s(0) p2 (r, δd )
X
Ad p2 (r, d),
(27)
1≤d≤n
where
p2 (r, d)=
√
Γ( n ) 2 π Γ( n−1 ) 2
R arccos( √d ) r
0
sinn−2 φ dφ, r >
√
d √ , 0, r ≤ d
(28)
which is a non-decreasing and continuous function of r such that p2 (0, d) = 0 and p2 (+∞, d) = 1/2. Therefore, X
fu (r) =
Pr{s}fu (r|s)
s
= fu (r|s(0) ) X Ad p2 (r, d) =
(29)
1≤d≤n
is also a non-decreasing and continuous function of r such that fu (0) = 0 and fu (+∞) ≥ 3/2. √ √ Furthermore, fu (r) is a strictly increasing function in the interval [ dmin, +∞) with fu ( dmin ) = 0. Hence there exists a unique r1 satisfying X
Ad p2 (r, d) = 1,
(30)
1≤d≤n
which is equivalent to that given in [1, (3.48)] by noticing that p2 (r, d) = 0 for d > r 2 . The parameterized SB for binary linear codes can be written as Z r1 Z +∞ Pr{E} ≤ fu (r)g(r) dr + g(r) dr 0
=
Z
r1
+∞
min{fu (r), 1}g(r) dr,
(31)
0
where g(r) and fu (r) are given in (21) and (29), respectively. The optimal parameter r1 is given by solving the equation (30), which does not depend on the SNR. It can be seen that (31) is
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!
s
Z 2
0
" d
!d ( z )
2
1
z1
sˆ
d1
d2
Z1
Fig. 2. The geometric interpretation of the TB and TSB for general codes.
exactly the sphere bound of Kasami et al [7][8]. It can also be proved that (31) is equivalent to that given in [1, (3.45)-(3.48)]. Firstly, we have shown that the optimal radius r1 satisfies (30), which is equivalent to that given in [1, (3.48)]. Secondly, by changing variables, z1 = r cos φ and y = r 2 , it can be verified that (31) is equivalent to that given in [1, Sec.3.2.5]. B. The Parameterized Tangential Bound In the derivation of the TB and TSB for binary codes, the equal-energy property plays a critical role. In the rest of this section, we show that the framework of the parameterized GFBT helps us to generalize the TB and TSB to general codes without the equal-energy property. The AWGN sample z can be separated by projection as a radial component zξ1 and n − 1 tangential (orthogonal) components {zξi , 2 ≤ i ≤ n}. Specifically, we set zξ1 to be the inner
product of z and −s/δd1 , where δd21 is the energy of s. When considering the pair-wise error
probability, we assume that zξ2 is the component that lies in the plane determined by s and sˆ. See Fig. 2 for reference. 1) Nested Regions: The parameterized TB chooses the nested regions to be a family of halfspaces Zξ1 ≤ zξ1 , where zξ1 ∈ R is the parameter. See Fig. 2 for reference. 2) Probability Density Function of the Parameter: The pdf of the parameter is 2
zξ 1 1 e− 2σ2 . g(zξ1 ) = √ 2πσ
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(32)
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3) Conditional Upper Bound: The parameterized TB chooses fu (zξ1 |s) to be the conditional union bound when transmitting the codeword s over the channel. Given that Zξ1 = zξ1 , the conditional pair-wise error probability is given by Z +∞ 2 zξ 1 − 22 2σ √ e dzξ2 , p2 (zξ1 , δd1 , δd2 , δd ) = 2πσ βd (zξ1 )
(33)
where δd − 2zξ1 cos θ , 2 sin θ
βd (zξ1 ) = and θ = arccos
δd21 + δd2 − δd22 2δd1 δd
(34)
.
(35)
Then the conditional union bound is given by X fu (zξ1 |s) = Bδd1 ,δd2 ,δd |s p2 (zξ1 , δd1 , δd2 , δd ).
(36)
4) The Parameterized TB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 .
(37)
δd1 ,δd2 ,δd
−∞
From (3), we define fu (zξ1 ) ,
X s
=
X
Pr{s}fu (zξ1 |s)
s
δd1 ,δd2 ,δd
X
=
X
Pr{s}
Bδd1 ,δd2 ,δd |s p2 (zξ1 , δd1 , δd2 , δd )
Bδd1 ,δd2 ,δd p2 (zξ1 , δd1 , δd2 , δd ).
(38)
δd1 ,δd2 ,δd
From (5), the parameterized TB for general codes can be written as X Pr{E} = Pr{s}Pr{E|s} s
≤
X
+∞
≤
Z
+∞
=
Z
Pr{s}
min
−∞
+∞
−∞
s
−∞
Z
( X s
min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 )
Pr{s}fu (zξ1 |s), 1 g(zξ1 ) dzξ1
min {fu (zξ1 ), 1} g(zξ1 ) dzξ1 ,
(39)
which is determined by the triangle Euclidean distance spectrum {Bδd1 ,δd2 ,δd }. May 11, 2014
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16
! (0)
s
Z2
0 d
sˆ
2 !d (z ) 1
n
z1
Z 1
Fig. 3. The geometric interpretation of the TB and TSB for binary linear codes.
5) Reduction to Binary Linear Codes: Similarly, for binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between √ a codeword sˆ with Hamming weight d and energy δd22 and s(0) with energy δd21 is δd = 2 d. √ Note that δd1 = δd2 = n, so Bδd ,δd ,δd |s(0) = Aδd |s(0) . See Fig. 3 for reference. Therefore, 1
2
from (9), (33) and (36), the conditional union bound fu (zξ1 |s(0) ) can be written as X Bδd ,δd ,δd |s(0) p2 (zξ1 , δd1 , δd2 , δd ) fu (zξ1 |s(0) ) = 1
2
δd1 ,δd2 ,δd
=
X δd
=
√ √ Aδd |s(0) p2 (zξ1 , n, n, δd )
X
Ad p2 (zξ1 , d),
(40)
1≤d≤n
where p2 (zξ1 , d) = and
May 11, 2014
Z
+∞ βd (zξ1 )
2
zξ 1 2 √ e− 2σ2 dzξ2 , 2πσ
√ √ d( n − zξ1 ) √ . βd (zξ1 ) = n−d
(41)
(42)
DRAFT
17
p2 (zξ1 , d) is a strictly increasing and continuous function of zξ1 such that p2 (−∞, d) = 0 and √ p2 ( n, d) = 1/2. Therefore, X Pr{s}fu (zξ1 |s) fu (zξ1 ) = s
= fu (zξ1 |s(0) ) X = Ad p2 (zξ1 , d)
(43)
1≤d≤n
√ is also a strictly increasing and continuous function of zξ1 such that fu (−∞) = 0 and fu ( n) ≥ √ 3/2. Hence there exists a unique solution zξ∗1 ≤ n satisfying n X
Ad p2 (zξ1 , d) = 1,
(44)
d=1
which is equivalent to that given in [1, (3.22)] by noticing that p2 (zξ1 , d) = Q d = δd2 /4.
√
√ d( n−zξ1 ) √ σ n−d
and
The parameterized TB for binary linear codes can be written as Z +∞ Z z∗ ξ1 g(zξ1 ) dzξ1 fu (zξ1 )g(zξ1 ) dzξ1 + Pr{E} ≤ zξ∗
−∞
=
Z
1
+∞
min{fu (zξ1 ), 1}g(zξ1 ) dzξ1 ,
(45)
−∞
where g(zξ1 ) and fu (zξ1 ) are given in (32) and (43), respectively. The optimal parameter zξ∗1 is given by solving the equation (44). It can be shown that (45) is equivalent to that given in [1, (3.21)]. C. The Parameterized Tangential-Sphere Bound Assume that n ≥ 3. 1) Nested Regions: Again, the parameterized TSB chooses the nested regions to be a family of half-spaces Zξ1 ≤ zξ1 , where zξ1 ∈ R is the parameter. 2) Probability Density Function of the Parameter: The pdf of the parameter is 2
zξ 1 1 e− 2σ2 . g(zξ1 ) = √ 2πσ
May 11, 2014
(46)
DRAFT
18
3) Conditional Upper Bound: Different from the parameterized TB, the parameterized TSB chooses fu (zξ1 |s) to be the conditional sphere bound when transmitting the codeword s over the channel. The conditional sphere bound given that Zξ1 = zξ1 can be derived as follows. Let R(r) be the (n−1)-dimensional sphere of radius r > 0 which is centered at (1−zξ1 /δd1 )s and located inside the hyper-plane Zξ1 = zξ1 . See Fig. 2 for reference. Given that the received vector y falls on the (n − 1)-dimensional sphere ∂R(r) in the hyperplane Zξ1 = zξ1 , the conditional pair-wise error probability is β (z ) ) R arccos( d r ξ1 ) Γ( n−1 2 √ sinn−3 φ dφ, r ≥ βd (zξ1 ), βd (zξ1 ) > 0 π Γ( n−2 ) 0 2 0, r < βd (zξ1 ), βd (zξ1 ) > 0 , p2 (zξ1 , r, δd1 , δd2 , δd ) = |βd (zξ )| n−1 1 ) Γ( 2 ) R arccos( n−3 r sin φ dφ, r ≥ |βd (zξ1 )|, βd (zξ1 ) ≤ 0 1 − √π Γ( n−2 ) 0 2 1, r < |βd (zξ1 )|, βd (zξ1 ) ≤ 0 (47) where
βd (zξ1 ) =
δd − 2zξ1 cos θ , 2 sin θ
and θ = arccos
δd21 + δd2 − δd22 2δd1 δd
(48)
.
From (24), we have the conditional sphere bound Z +∞ fu (zξ1 |s) = min {fs (zξ1 , r|s), 1} gs (r) dr,
(49)
(50)
0
where
r2
gs (r) =
2r n−2e− 2σ2 2
n−1 2
σ n−1 Γ( n−1 ) 2
, r ≥ 0,
(51)
and fs (zξ1 , r|s) =
X
δd1 ,δd2 ,δd
Bδd1 ,δd2 ,δd |s p2 (zξ1 , r, δd1 , δd2 , δd ).
4) The Parameterized TSB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 −∞ Z +∞ Z +∞ ≤ min min {fs (zξ1 , r|s), 1} gs (r) dr, 1 g(zξ1 ) dzξ1 . −∞
May 11, 2014
(52)
0
(53)
DRAFT
19
From (3), we define X
fs (zξ1 , r) ,
Pr{s}fs (zξ1 , r|s)
s
X
=
s
δd1 ,δd2 ,δd
X
=
X
Pr{s}
Bδd1 ,δd2 ,δd |s p2 (zξ1 , r, δd1 , δd2 , δd )
Bδd1 ,δd2 ,δd p2 (zξ1 , r, δd1 , δd2 , δd ).
(54)
δd1 ,δd2 ,δd
From (5), the parameterized TSB for general codes can be written as X Pr{E} = Pr{s}Pr{E|s} s
≤
X
+∞
≤
Z
+∞
=
Z
Pr{s}
+∞
min
−∞
s
min
−∞
−∞
Z
(Z
+∞
Z
+∞
Z
min
0
min
+∞
min {fs (zξ1 , r|s), 1} gs (r) dr, 1 g(zξ1 ) dzξ1
0
( X s
)
)
Pr {s} fs (zξ1 , r|s), 1 gs (r) dr, 1 g(zξ1 ) dzξ1
min {fs (zξ1 , r), 1} gs (r) dr, 1 g(zξ1 ) dzξ1 ,
0
(55)
which is determined by the triangle Euclidean distance spectrum {Bδd1 ,δd2 ,δd }. 5) Reduction to Binary Linear Codes: Similarly, for binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between a √ codeword sˆ with Hamming weight d and energy δd22 and s(0) with energy δd21 is δd = 2 d. Note √ that δd1 = δd2 = n, so Bδd ,δd ,δd |s(0) = Aδd |s(0) . See Fig. 3 for reference. Therefore, from (50), 1
2
the conditional sphere bound fu (zξ1 |s(0) ) can be written as Z +∞ (0) fu (zξ1 |s ) = min fs (zξ1 , r|s(0) ), 1 gs (r) dr.
(56)
0
From (9), (47) and (52), we have
fs (zξ1 , r|s(0) ) =
X
Bδd
δd1 ,δd2 ,δd
=
X δd
=
1
,δd2 ,δd |s(0) p2 (zξ1 , r, δd1 , δd2 , δd )
Aδd |s(0) p2 (zξ1 , r,
X
√
Ad p2 (zξ1 , r, d),
n,
√
n, δd ) (57)
1≤d≤n
May 11, 2014
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20
where
p2 (zξ1 , r, d) =
√
1−
and
Γ( n−1 ) 2 π Γ( n−2 ) 2
Γ( n−1 ) 2 √ π Γ( n−2 ) 2
R arccos( βd (zξ1 ) ) r
0
R
|βd (zξ )| 1 ) arccos( r
0
fu (zξ1 ) =
X s
√
√
n
√
n √ , (58) n−3 sin φ dφ, r ≥ |βd (zξ1 )|, zξ1 ≥ n √ 1, r < |βd (zξ1 )|, zξ1 ≥ n 0, r < βd (zξ1 ), zξ1
Upper Bounds On the ML Decoding Error Probability of General Codes over AWGN
arXiv:1308.3303v2 [cs.IT] 26 Dec 2013
Channels Qiutao Zhuang, Jia Liu, and Xiao Ma, Member, IEEE
Abstract In this paper, parameterized Gallager’s first bounding technique (GFBT) is presented by introducing nested Gallager regions, to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The three well-known bounds, namely, the sphere bound (SB) of Herzberg and Poltyrev, the tangential bound (TB) of Berlekamp, and the tangential-sphere bound (TSB) of Poltyrev, are generalized to general codes without the properties of geometrical uniformity and equal energy. When applied to the binary linear codes, the three generalized bounds are reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al., which was rarely cited in the literatures.
Index Terms Additive white Gaussian noise (AWGN) channel, Gallager’s first bounding technique (GFBT), general codes, maximum-likelihood (ML) decoding, parameterized GFBT, trellis code.
I. I NTRODUCTION In most scenarios, there do not exist easy ways to compute the exact decoding error probabilities for specific codes and ensembles. Therefore, deriving tight analytical bounds is an important research subject in the field of coding theory and practice. Since the early 1990s, spurred by the successes of the near-capacity-achieving codes, renewed attentions have been paid to the The authors are with the Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou 510006, China. J. Liu is also with the College of Information Science and Technology, Zhongkai University of Agriculture and Engineering, Guangzhou 510225, China. (Email: [email protected]) May 11, 2014
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2
performance analysis of the maximum-likelihood (ML) decoding algorithm. Though the ML decoding algorithm is prohibitively complex for most practical codes, tight bounds can be used to predict their performance without resorting to computer simulations. As mentioned in [1], most bounding techniques have connections to either the 1965 Gallager bound [2–5] or the 1961 Gallager bound [6–18] based on Gallager’s first bounding technique (GFBT). However, most previously reported upper bounds are focusing on binary linear codes. For binary linear codes modulated by binary phase shift keying (BPSK), there are two main properties, which are geometrical uniformity and equal energy. The geometrical uniformity allows us to make an assumption that the all-zero codeword is the transmitted one, while the property of equal energy is critical to derive the tangential bound (TB) [6] and the tangential-sphere bound (TSB) [10]. For general codes without these two properties, performance analysis becomes more difficult than that for binary linear codes. In this paper, we present parameterized GFBT by introducing nested Gallager regions to derive upper bounds on the ML decoding error probability of general codes over AWGN channels. The main contributions as well as the structure of this paper are summarized as follows. 1) We present in Sec. II the parameterized GFBT for general codes. We also present a necessary and sufficient condition on the optimal parameter, and a sufficient condition (with a simple geometrical explanation) under which the optimal parameter does not depend on the signalto-noise ratio (SNR). 2) Within the general framework based on the introduced nested Gallager regions, three existing upper bounds, the sphere bound (SB) of Herzberg and Poltyrev [9], the tangential bound (TB) of Berlekamp [6] and the tangential-sphere bound (TSB) of Poltyrev [10], are generalized in Sec. III to general codes without the properties of geometrical uniformity and equal energy. The three upper bounds are then applied to binary linear codes and reduced to the conventional ones. The new derivation also reveals that the SB of Herzberg and Poltyrev is equivalent to the SB of Kasami et al. [7] [8], which was rarely cited in the literatures. 3) We use in Sec. IV terminated trellis codes [19] to illustrate how to calculate the parameterized Gallager first bounds on the frame-error probability. Numerical results are also presented in Sec. IV. Sec. V concludes this paper.
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3
II. T HE PARAMETERIZED G ALLAGER ’ S F IRST B OUNDS A. General Codes A general code C(n, M) ⊂ Rn , in this paper, means a set that contains M n-dimensional real vectors (referred to as codewords). The squared Euclidean distance between a codeword s and the origin point O of the n-dimensional space, denoted by ksk2 , is also referred to as the energy of this codeword. If all codewords have the same energy, we say that the code has the property of equal energy. Given a codeword s, we denote Aδd |s the number of codewords having the Euclidean distance δd with s. We define ∆
Aδd =
X s
Pr{s}Aδd |s ,
(1)
which is the average number of ordered pairs of codewords with Euclidean distance δd . Definition 1: The Euclidean distance enumerating function of a general code C(n, M) is defined as ∆
A(X) =
X
2
Aδd X δd ,
(2)
δd
where X is a dummy variable and the summation is over all possible distance δd . For a general code, there exist at most M2 non-zero coefficients {Aδd }, which is referred to as the Euclidean distance spectrum.
To derive tangential bounds, we also need another distance spectrum for general codes. Given a codeword s with energy δd21 , we denote Bδd1 ,δd2 ,δd |s the number of codewords sˆ having energy δd22 and the Euclidean distance δd with s. We define X ∆ Bδd1 ,δd2 ,δd = Pr{s}Bδd1 ,δd2 ,δd |s ,
(3)
s
which is the average number of ordered pairs of codewords with the Euclidean distance δd and energies δd21 and δd22 , respectively. Definition 2: The triangle Euclidean distance enumerating function of a general code C(n, M) is defined as ∆
B(X, Y, Z) =
X
2
2
2
Bδd1 ,δd2 ,δd X δd1 Y δd2 Z δd ,
(4)
δd1 ,δd2 ,δd
where X, Y, Z are three dummy variables. We call {Bδd1 ,δd2 ,δd } the triangle Euclidean distance spectrum of the given code. May 11, 2014
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4
B. The Conventional Union Bound Suppose that a codeword s = (s0 , s1 , · · · , sn−1 ) ∈ C(n, M) is transmitted over an AWGN channel. Let y = s+z be the received vector, where z is a vector of independent Gaussian random variables with zero mean and variance σ 2 . For AWGN channels, the maximum-likelihood (ML) decoding is equivalent to finding the nearest codeword ˆs to y. The decoding error probability Pr{E} is Pr{E} =
X
Pr{s}Pr{E|s},
(5)
s
where Pr{E|s} is the conditional decoding error probability when transmitting s over the channel. As usual, we assume that each codeword s is transmitted with equal probability, that is Pr{s} = 1/M. With this assumption, the code rate is is
P
s
ksk2
nM σ2
log M n
and the signal-to-noise ratio (SNR)
.
The conventional union bound on the ML decoding error probability of a general code C(n, M) is Pr{E} =
X
Pr{s}Pr{E|s}
s
δd Aδd |s Q Pr{s} ≤ 2σ s δd XX δd Pr{s}Aδd |s Q = 2σ s δd X δd Aδd Q = , 2σ δ X
X
(6)
d
where Q
δd 2σ
is the pair-wise error probability with Z +∞ z2 1 ∆ √ e− 2 dz. Q(x) = 2π x
(7)
The union bound is simple since it involves only the Q-function and does not require the code structure other than the Euclidean distance spectrum. However, the union bound is loose and even diverges in the low-SNR region. One way to solve this issue is to use the GFBT / R|s}, Pr{E|s} ≤ Pr{E, y ∈ R|s} + Pr{y ∈
(8)
where E denotes the conditional error event, y denotes the received signal vector, and R denotes an arbitrary region around the transmitted signal vector s. The first term in the right hand May 11, 2014
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5
side (RHS) of (8) is usually bounded by the conditional union bound, while the second term in the RHS of (8) represents the probability of the event that the received vector y falls outside the region R, which is considered to be decoded incorrectly even if it may not fall outside the Voronoi region [20] [21] of the transmitted codeword. For convenience, we call (8) R-bound. Intuitively, the more similar the region R is to the Voronoi region of the transmitted signal vector, the tighter the R-bound is. Therefore, both the shape and the size of the region R are critical to GFBT. Given the region’s shape, one can optimize its size to obtain the tightest R-bound. Different from most existing works, where the size of R is optimized by setting to be zero the partial derivative of the bound with respect to a parameter (specifying the size), we will propose an alternative method by introducing nested Gallager’s regions in the subsection II-D. C. Binary Linear Codes For a binary linear block code C(n, M) of dimension k = log2 M, length n, and minimum Hamming distance dmin, suppose that a codeword c is modulated by binary phase shift keying (BPSK), resulting in a bipolar signal vector s with st = 1−2ct for 0 ≤ t ≤ n−1. Without loss of generality, we assume that the code C has at least three non-zero codewords, i.e., its dimension
k > 1, and the transmitted codeword is the all-zero codeword c(0) (with bipolar image s(0) ).
Let cˆ (with bipolar image sˆ) be a codeword of Hamming weight d, then the Euclidean distance √ between s(0) and sˆ is δd = 2 d. We define ∆
Ad = Aδd |s(0) ,
(9)
which is the number of codewords with Hamming weight d. Since the constellation of binary linear block code is geometrically uniform and each codeword is assumed to be transmitted with equal probability, we have Pr{E} =
X
Pr{s}Pr{E|s}
s
= Pr{E|s(0) } √ ! X d , ≤ Ad Q σ d
(10)
where {Ad } is the weight distribution of the code C.
May 11, 2014
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6
D. GFBT with Parameters In this subsection, we will present parameterized GFBT by introducing nested Gallager regions with parameters so that Gallager bounds can be extended to general codes conveniently. To this end, let {R(r), r ∈ I ⊆ R} be a family of Gallager’s regions with the same shape and parameterized by r ∈ I. For example, the nested regions can be chosen as a family of ndimensional spheres of radius r ≥ 0 centered at the transmitted codeword s. We make the following assumptions. Assumptions. A1. The regions {R(r), r ∈ I ⊆ R} are nested and their boundaries partition the whole space Rn . That is,
R(r1 ) ⊂ R(r2 ) if r1 < r2 , ∂R(r1 ) and
\
(11)
∂R(r2 ) = ∅ if r1 6= r2 ,
Rn =
[
∂R(r),
(12)
(13)
r∈I
where ∂R(r) denotes the boundary surface of the region R(r). A2. Define a functional R : y 7→ r whenever y ∈ ∂R(r). The randomness of the received vector y then induces a random variable R. We assume that R has a probability density function (pdf) g(r). A3. We also assume that Pr{E|y ∈ ∂R(r), s} can be upper-bounded by a computable upper bound fu (r|s). For ease of notation, we may enlarge the index set I to R by setting g(r) ≡ 0 for r ∈ / I.
Under the above assumptions, we have the following parameterized GFBT 1 . Proposition 1: For any r ∗ ∈ R, Pr{E|s} ≤ 1
Z
r∗
fu (r|s)g(r) dr + −∞
Z
+∞
g(r) dr.
(14)
r∗
Strictly speaking, we need one more assumption that fu (r|s) is measurable with respect to g(r).
May 11, 2014
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7
Proof: Pr{E|s} = Pr{E, y ∈ R(r ∗ )|s} + Pr{E, y ∈ / R(r ∗ )|s} / R(r ∗ )|s} ≤ Pr{E, y ∈ R(r ∗ )|s} + Pr{y ∈ Z r∗ Z +∞ fu (r|s)g(r) dr + ≤ g(r) dr. −∞
r∗
An immediate question is how to choose r ∗ to make the above bound as tight as possible? A natural method is to set the derivative of (14) with respect to r ∗ to be zero and then solve the equation. In this paper, we propose an alternative method for gaining insight into the optimal parameter. Before presenting a necessary and sufficient condition on the optimal parameter, we need emphasize that the computable bound fu (r|s) may exceed one. We also assume that fu (r|s) is non-trivial, i.e., there exists some r such that fu (r|s) ≤ 1. For example, fu (r|s) can be taken as the union bound conditional on y ∈ ∂R(r). Theorem 1: Assume that fu (r|s) is a non-decreasing and continuous function of r. Let r1 be a parameter that minimizes the upper bound as shown in (14). Then r1 = sup{r ∈ I} if fu (r|s) < 1 for all r ∈ I; otherwise, r1 can be taken as any solution of fu (r|s) = 1. Furthermore, if fu (r|s) is strictly increasing in an interval [rmin , rmax ] such that fu (rmin |s) < 1 and fu (rmax |s) > 1, there exists a unique r1 ∈ [rmin , rmax ] such that fu (r1 |s) = 1. Proof: The second part is obvious since the function fu (r|s) is strictly increasing and continuous, which is helpful for solving numerically the equation fu (r|s) = 1. To prove the first part, it suffices to prove that neither r0 < sup{r ∈ I} with fu (r0 |s) < 1 nor r2 with fu (r2 |s) > 1 can be optimal. Let r0 < sup{r ∈ I} such that fu (r0 |s) < 1. Since fu (r|s) is continuous and r0 < sup{r ∈ I},
we can find I ∋ r ′ > r0 such that fu (r ′ |s) < 1. Then we have
May 11, 2014
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8
Z
r0
fu (r|s)g(r) dr +
−∞
=
Z
r0
fu (r|s)g(r) dr +
−∞
>
Z
r0
fu (r|s)g(r) dr +
−∞
=
Z
r′
fu (r|s)g(r) dr +
−∞
Z Z Z Z
+∞
g(r) dr r0 r′
g(r) dr +
+∞
Z
g(r) dr
r′
r0 r′
fu (r|s)g(r) dr +
+∞
Z
g(r) dr
r′
r0 +∞
g(r) dr, r′
where we have used the fact that fu (r|s) < 1 for r ∈ [r0 , r ′ ]. This shows that r ′ is better than r0 . Suppose that r2 is a parameter such that fu (r2 |s) > 1. Since fu (r|s) is continuous and nontrivial, we can find r1 < r2 such that fu (r1 |s) = 1. Then we have Z r2 Z +∞ fu (r|s)g(r) dr + g(r) dr −∞ r1
= > =
Z
−∞ Z r1
−∞ Z r1
r2 r2
fu (r|s)g(r) dr + fu (r|s)g(r) dr + fu (r|s)g(r) dr +
Z
r1 Z r2
fu (r|s)g(r) dr +
Z
+∞
g(r) dr
r2
g(r) dr +
r1 Z +∞
Z
+∞
g(r) dr
r2
g(r) dr,
r1
−∞
where we have used a condition that fu (r|s) > 1 for r ∈ (r1 , r2 ], which can be fulfilled by choosing r1 to be the maximum solution of fu (r|s) = 1. This shows that r1 is better than r2 . Corollary 1: Let fu (r|s) be a non-decreasing and continuous function of r. If fu (r|s) does not depend on the SNR, then the optimal parameter r1 minimizing the upper bound (14) does not depend on the SNR, either. Proof: It is an immediate result from Theorem 1. Theorem 1 requires fu (r|s) to be a non-decreasing and continuous function of r, which can be fulfilled for several well-known bounds. Without such a condition, we may use the following more general theorem. Theorem 2: For any measurable subset A ⊂ I, we have Z Z Pr{E|s} ≤ fu (r|s)g(r) dr + r∈A
May 11, 2014
g(r) dr.
(15)
r ∈A /
DRAFT
9
Within this type, the tightest bound is Z Pr{E|s} ≤
fu (r|s)g(r) dr +
r∈I0
Z
g(r) dr,
(16)
r ∈I / 0
where I0 = {r ∈ I|fu (r|s) < 1}. Equivalently, we have Z min{fu (r|s), 1}g(r) dr. Pr{E|s} ≤
(17)
r∈I
Proof: Let G =
S
r∈A
∂R(r), we have
/ G|s} Pr{E|s} ≤ Pr{E, y ∈ G|s} + Pr{y ∈ Z Z = fu (r|s)g(r) dr + g(r) dr. r∈A
r ∈A /
/ Define A0 = {r ∈ A|fu (r|s) < 1} and A1 = {r ∈ A|fu (r|s) ≥ 1}. Similarly, define B0 = {r ∈ A|fu (r|s) < 1} and B1 = {r ∈ / A|fu (r|s) ≥ 1}. Noticing that Z Z Z fu (r|s)g(r) dr + g(r) dr fu (r|s)g(r) dr ≥ r∈A0 r∈A1 r∈A Z Z Z g(r) dr, fu (r|s)g(r) dr + g(r) dr ≥ r ∈A /
r∈B1
r∈B0
we have Z
≥ = =
Z
Z
Z
fu (r|s)g(r) dr + r∈A
Z
r ∈A /
r∈A0
S
fu (r|s)g(r) dr + B0
fu (r|s)g(r) dr + r∈I0
g(r) dr Z r∈A1
Z
S
g(r) dr
B1
g(r) dr
r ∈I / 0
min{fu (r|s), 1}g(r) dr. r∈I
E. Conditional Pair-Wise Error Probabilities Let δd denote the Euclidean distance between s (the transmitted codeword) and a codeword sˆ. The pair-wise error probability conditional on the event {y ∈ ∂R(r)}, denoted by p2 (r, δd ), is
May 11, 2014
p2 (r, δd ) = Pr ky − sˆk ≤ ky − sk | y ∈ ∂R(r) R f (y) dy sk≤ky−sk, y∈∂R(r) ky−ˆ R , = f (y) dy y∈∂R(r)
(18) DRAFT
10
where f (y) is the pdf of y. Noticing that, different from the unconditional pair-wise error probabilities, p2 (r, δd ) may be zero for some r. We have the following lemma. Lemma 1: Suppose that, conditional on y ∈ ∂R(r), the received vector y is uniformly distributed over ∂R(r). Then the conditional pair-wise error probability p2 (r, δd ) does not depend on the SNR. Proof: Since f (y) is constant for y ∈ ∂R(r), we have, by canceling f (y) from both the numerator and the denominator of (18), p2 (r, δd ) =
R
sk≤ky−sk,y∈∂R(r) ky−ˆ
R
y∈∂R(r)
dy
dy ,
(19)
which shows that the conditional pair-wise error probability can be represented as a ratio of two “surface area” and hence does not depend on the SNR. Theorem 3: Let fu (r|s) be the conditional union bound, that is, X Aδd |s p2 (r, δd ). fu (r|s) =
(20)
δd
Suppose that, conditional on y ∈ ∂R(r), the received vector y is uniformly distributed over ∂R(r). If fu (r|s) is a non-decreasing and continuous function of r, then the optimal parameter r1 minimizing the bound (14) does not depend on SNR but only on the distance spectrum of the code. Proof: From Lemma 1, we know that fu (r|s) does not depend on the SNR. From Corollary 1, we know that r1 does not depend on the SNR. More generally, without the condition that fu (r|s) is a non-decreasing and continuous function of r, the optimal interval I0 defined in Theorem 2 does not depend on the SNR, either.
F. General Framework of Parameterized GFBT From the above subsection, we can see that there are three main steps to derive a parameterized GFBT. First, choose properly nested regions specified by a parameter. Second, find the pdf of the parameter. Finally, find a computable upper bound on the conditional decoding error probability given that the received vector falls on the boundary of a parameter-specified region. The key of the third step is to find the “projection” of the codewords to the boundary. Here, the “projection” May 11, 2014
DRAFT
11
sˆ
r
d
2
s
Fig. 1. The geometric interpretation of the SB for general codes.
means that the intersection between the perpendicular bisector of the segment s sˆ (s and sˆ are the transmitted codeword and decoded codeword, respectively) and the boundary. III. S INGLE -PARAMETERIZED U PPER B OUNDS
FOR
G ENERAL C ODES
For a general code, the property of geometrical uniformity may not hold. As a result, we can not assume a particular transmitted codeword and must average over all conditional error probabilities. In this section, we will first derive the conditional upper bound of Pr{E|s} when transmitting the codeword s over the channel according to the framework of the parameterized GFBT, and then obtain the upper bound of Pr{E} from (5). A. The Parameterized Sphere Bound 1) Nested Regions: The parameterized SB chooses the nested regions to be a family of ndimensional spheres centered at the transmitted signal vector s, that is, R(r) = {y | ky−sk ≤ r}, where r ≥ 0 is the parameter. See Fig. 1 for reference. 2) Probability Density Function of the Parameter: The pdf of the parameter is r2
2r n−1 e− 2σ2 g(r) = n n n , r ≥ 0. 2 2 σ Γ( 2 )
(21)
3) Conditional Upper Bound: The parameterized SB chooses fu (r|s) to be the conditional union bound when transmitting the codeword s over the channel. Given that ||y − s|| = r, y is uniformly distributed over ∂R(r). Hence the conditional pair-wise error probability p2 (r, δd ) May 11, 2014
DRAFT
12
does not depend on the SNR and can be evaluated as the ratio of the surface area of a spherical cap to that of the whole sphere. That is, n R arccos( δd ) n−2 ) √ Γ( 2n−1 2r sin φ dφ, r > π Γ( 2 ) 0 p2 (r, δd )= 0, r ≤
δd 2 δd 2
.
Then the conditional union bound is given by X fu (r|s) = Aδd |s p2 (r, δd ).
(22)
(23)
δd
4) The Parameterized SB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (r|s), 1}g(r) dr.
(24)
0
From (1), we define fu (r) ,
X
Pr{s}fu (r|s)
s
=
X
Pr{s}
s
=
δd
=
δd
XX X
X
s
Aδd |s p2 (r, δd )
Pr{s}Aδd |s p2 (r, δd )
Aδd p2 (r, δd ).
(25)
δd
From (5), the parameterized SB for general codes can be written as X Pr{s}Pr{E|s} Pr{E} = s
≤
X
≤
Z
=
Z
Pr{s}
+∞
min +∞ 0
+∞
min {fu (r|s), 1} g(r) dr
0
s
0
Z
(
X
)
Pr{s}fu (r|s), 1 g(r) dr
s
min {fu (r), 1} g(r) dr,
(26)
which is determined by the Euclidean distance spectrum {Aδd }.
May 11, 2014
DRAFT
13
5) Reduction to Binary Linear Codes: For binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between a codeword sˆ √ with Hamming weight d and s(0) is δd = 2 d. Therefore, from (9), (22) and (23), the conditional union bound fu (r|s(0) ) can be written as fu (r|s(0) ) =
X δd
=
Aδd |s(0) p2 (r, δd )
X
Ad p2 (r, d),
(27)
1≤d≤n
where
p2 (r, d)=
√
Γ( n ) 2 π Γ( n−1 ) 2
R arccos( √d ) r
0
sinn−2 φ dφ, r >
√
d √ , 0, r ≤ d
(28)
which is a non-decreasing and continuous function of r such that p2 (0, d) = 0 and p2 (+∞, d) = 1/2. Therefore, X
fu (r) =
Pr{s}fu (r|s)
s
= fu (r|s(0) ) X Ad p2 (r, d) =
(29)
1≤d≤n
is also a non-decreasing and continuous function of r such that fu (0) = 0 and fu (+∞) ≥ 3/2. √ √ Furthermore, fu (r) is a strictly increasing function in the interval [ dmin, +∞) with fu ( dmin ) = 0. Hence there exists a unique r1 satisfying X
Ad p2 (r, d) = 1,
(30)
1≤d≤n
which is equivalent to that given in [1, (3.48)] by noticing that p2 (r, d) = 0 for d > r 2 . The parameterized SB for binary linear codes can be written as Z r1 Z +∞ Pr{E} ≤ fu (r)g(r) dr + g(r) dr 0
=
Z
r1
+∞
min{fu (r), 1}g(r) dr,
(31)
0
where g(r) and fu (r) are given in (21) and (29), respectively. The optimal parameter r1 is given by solving the equation (30), which does not depend on the SNR. It can be seen that (31) is
May 11, 2014
DRAFT
14
!
s
Z 2
0
" d
!d ( z )
2
1
z1
sˆ
d1
d2
Z1
Fig. 2. The geometric interpretation of the TB and TSB for general codes.
exactly the sphere bound of Kasami et al [7][8]. It can also be proved that (31) is equivalent to that given in [1, (3.45)-(3.48)]. Firstly, we have shown that the optimal radius r1 satisfies (30), which is equivalent to that given in [1, (3.48)]. Secondly, by changing variables, z1 = r cos φ and y = r 2 , it can be verified that (31) is equivalent to that given in [1, Sec.3.2.5]. B. The Parameterized Tangential Bound In the derivation of the TB and TSB for binary codes, the equal-energy property plays a critical role. In the rest of this section, we show that the framework of the parameterized GFBT helps us to generalize the TB and TSB to general codes without the equal-energy property. The AWGN sample z can be separated by projection as a radial component zξ1 and n − 1 tangential (orthogonal) components {zξi , 2 ≤ i ≤ n}. Specifically, we set zξ1 to be the inner
product of z and −s/δd1 , where δd21 is the energy of s. When considering the pair-wise error
probability, we assume that zξ2 is the component that lies in the plane determined by s and sˆ. See Fig. 2 for reference. 1) Nested Regions: The parameterized TB chooses the nested regions to be a family of halfspaces Zξ1 ≤ zξ1 , where zξ1 ∈ R is the parameter. See Fig. 2 for reference. 2) Probability Density Function of the Parameter: The pdf of the parameter is 2
zξ 1 1 e− 2σ2 . g(zξ1 ) = √ 2πσ
May 11, 2014
(32)
DRAFT
15
3) Conditional Upper Bound: The parameterized TB chooses fu (zξ1 |s) to be the conditional union bound when transmitting the codeword s over the channel. Given that Zξ1 = zξ1 , the conditional pair-wise error probability is given by Z +∞ 2 zξ 1 − 22 2σ √ e dzξ2 , p2 (zξ1 , δd1 , δd2 , δd ) = 2πσ βd (zξ1 )
(33)
where δd − 2zξ1 cos θ , 2 sin θ
βd (zξ1 ) = and θ = arccos
δd21 + δd2 − δd22 2δd1 δd
(34)
.
(35)
Then the conditional union bound is given by X fu (zξ1 |s) = Bδd1 ,δd2 ,δd |s p2 (zξ1 , δd1 , δd2 , δd ).
(36)
4) The Parameterized TB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 .
(37)
δd1 ,δd2 ,δd
−∞
From (3), we define fu (zξ1 ) ,
X s
=
X
Pr{s}fu (zξ1 |s)
s
δd1 ,δd2 ,δd
X
=
X
Pr{s}
Bδd1 ,δd2 ,δd |s p2 (zξ1 , δd1 , δd2 , δd )
Bδd1 ,δd2 ,δd p2 (zξ1 , δd1 , δd2 , δd ).
(38)
δd1 ,δd2 ,δd
From (5), the parameterized TB for general codes can be written as X Pr{E} = Pr{s}Pr{E|s} s
≤
X
+∞
≤
Z
+∞
=
Z
Pr{s}
min
−∞
+∞
−∞
s
−∞
Z
( X s
min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 )
Pr{s}fu (zξ1 |s), 1 g(zξ1 ) dzξ1
min {fu (zξ1 ), 1} g(zξ1 ) dzξ1 ,
(39)
which is determined by the triangle Euclidean distance spectrum {Bδd1 ,δd2 ,δd }. May 11, 2014
DRAFT
16
! (0)
s
Z2
0 d
sˆ
2 !d (z ) 1
n
z1
Z 1
Fig. 3. The geometric interpretation of the TB and TSB for binary linear codes.
5) Reduction to Binary Linear Codes: Similarly, for binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between √ a codeword sˆ with Hamming weight d and energy δd22 and s(0) with energy δd21 is δd = 2 d. √ Note that δd1 = δd2 = n, so Bδd ,δd ,δd |s(0) = Aδd |s(0) . See Fig. 3 for reference. Therefore, 1
2
from (9), (33) and (36), the conditional union bound fu (zξ1 |s(0) ) can be written as X Bδd ,δd ,δd |s(0) p2 (zξ1 , δd1 , δd2 , δd ) fu (zξ1 |s(0) ) = 1
2
δd1 ,δd2 ,δd
=
X δd
=
√ √ Aδd |s(0) p2 (zξ1 , n, n, δd )
X
Ad p2 (zξ1 , d),
(40)
1≤d≤n
where p2 (zξ1 , d) = and
May 11, 2014
Z
+∞ βd (zξ1 )
2
zξ 1 2 √ e− 2σ2 dzξ2 , 2πσ
√ √ d( n − zξ1 ) √ . βd (zξ1 ) = n−d
(41)
(42)
DRAFT
17
p2 (zξ1 , d) is a strictly increasing and continuous function of zξ1 such that p2 (−∞, d) = 0 and √ p2 ( n, d) = 1/2. Therefore, X Pr{s}fu (zξ1 |s) fu (zξ1 ) = s
= fu (zξ1 |s(0) ) X = Ad p2 (zξ1 , d)
(43)
1≤d≤n
√ is also a strictly increasing and continuous function of zξ1 such that fu (−∞) = 0 and fu ( n) ≥ √ 3/2. Hence there exists a unique solution zξ∗1 ≤ n satisfying n X
Ad p2 (zξ1 , d) = 1,
(44)
d=1
which is equivalent to that given in [1, (3.22)] by noticing that p2 (zξ1 , d) = Q d = δd2 /4.
√
√ d( n−zξ1 ) √ σ n−d
and
The parameterized TB for binary linear codes can be written as Z +∞ Z z∗ ξ1 g(zξ1 ) dzξ1 fu (zξ1 )g(zξ1 ) dzξ1 + Pr{E} ≤ zξ∗
−∞
=
Z
1
+∞
min{fu (zξ1 ), 1}g(zξ1 ) dzξ1 ,
(45)
−∞
where g(zξ1 ) and fu (zξ1 ) are given in (32) and (43), respectively. The optimal parameter zξ∗1 is given by solving the equation (44). It can be shown that (45) is equivalent to that given in [1, (3.21)]. C. The Parameterized Tangential-Sphere Bound Assume that n ≥ 3. 1) Nested Regions: Again, the parameterized TSB chooses the nested regions to be a family of half-spaces Zξ1 ≤ zξ1 , where zξ1 ∈ R is the parameter. 2) Probability Density Function of the Parameter: The pdf of the parameter is 2
zξ 1 1 e− 2σ2 . g(zξ1 ) = √ 2πσ
May 11, 2014
(46)
DRAFT
18
3) Conditional Upper Bound: Different from the parameterized TB, the parameterized TSB chooses fu (zξ1 |s) to be the conditional sphere bound when transmitting the codeword s over the channel. The conditional sphere bound given that Zξ1 = zξ1 can be derived as follows. Let R(r) be the (n−1)-dimensional sphere of radius r > 0 which is centered at (1−zξ1 /δd1 )s and located inside the hyper-plane Zξ1 = zξ1 . See Fig. 2 for reference. Given that the received vector y falls on the (n − 1)-dimensional sphere ∂R(r) in the hyperplane Zξ1 = zξ1 , the conditional pair-wise error probability is β (z ) ) R arccos( d r ξ1 ) Γ( n−1 2 √ sinn−3 φ dφ, r ≥ βd (zξ1 ), βd (zξ1 ) > 0 π Γ( n−2 ) 0 2 0, r < βd (zξ1 ), βd (zξ1 ) > 0 , p2 (zξ1 , r, δd1 , δd2 , δd ) = |βd (zξ )| n−1 1 ) Γ( 2 ) R arccos( n−3 r sin φ dφ, r ≥ |βd (zξ1 )|, βd (zξ1 ) ≤ 0 1 − √π Γ( n−2 ) 0 2 1, r < |βd (zξ1 )|, βd (zξ1 ) ≤ 0 (47) where
βd (zξ1 ) =
δd − 2zξ1 cos θ , 2 sin θ
and θ = arccos
δd21 + δd2 − δd22 2δd1 δd
(48)
.
From (24), we have the conditional sphere bound Z +∞ fu (zξ1 |s) = min {fs (zξ1 , r|s), 1} gs (r) dr,
(49)
(50)
0
where
r2
gs (r) =
2r n−2e− 2σ2 2
n−1 2
σ n−1 Γ( n−1 ) 2
, r ≥ 0,
(51)
and fs (zξ1 , r|s) =
X
δd1 ,δd2 ,δd
Bδd1 ,δd2 ,δd |s p2 (zξ1 , r, δd1 , δd2 , δd ).
4) The Parameterized TSB: From (17), we have Z +∞ Pr{E|s} ≤ min{fu (zξ1 |s), 1}g(zξ1 ) dzξ1 −∞ Z +∞ Z +∞ ≤ min min {fs (zξ1 , r|s), 1} gs (r) dr, 1 g(zξ1 ) dzξ1 . −∞
May 11, 2014
(52)
0
(53)
DRAFT
19
From (3), we define X
fs (zξ1 , r) ,
Pr{s}fs (zξ1 , r|s)
s
X
=
s
δd1 ,δd2 ,δd
X
=
X
Pr{s}
Bδd1 ,δd2 ,δd |s p2 (zξ1 , r, δd1 , δd2 , δd )
Bδd1 ,δd2 ,δd p2 (zξ1 , r, δd1 , δd2 , δd ).
(54)
δd1 ,δd2 ,δd
From (5), the parameterized TSB for general codes can be written as X Pr{E} = Pr{s}Pr{E|s} s
≤
X
+∞
≤
Z
+∞
=
Z
Pr{s}
+∞
min
−∞
s
min
−∞
−∞
Z
(Z
+∞
Z
+∞
Z
min
0
min
+∞
min {fs (zξ1 , r|s), 1} gs (r) dr, 1 g(zξ1 ) dzξ1
0
( X s
)
)
Pr {s} fs (zξ1 , r|s), 1 gs (r) dr, 1 g(zξ1 ) dzξ1
min {fs (zξ1 , r), 1} gs (r) dr, 1 g(zξ1 ) dzξ1 ,
0
(55)
which is determined by the triangle Euclidean distance spectrum {Bδd1 ,δd2 ,δd }. 5) Reduction to Binary Linear Codes: Similarly, for binary linear codes, the transmitted codeword s can be assumed to be the all-zero codeword s(0) . The Euclidean distance between a √ codeword sˆ with Hamming weight d and energy δd22 and s(0) with energy δd21 is δd = 2 d. Note √ that δd1 = δd2 = n, so Bδd ,δd ,δd |s(0) = Aδd |s(0) . See Fig. 3 for reference. Therefore, from (50), 1
2
the conditional sphere bound fu (zξ1 |s(0) ) can be written as Z +∞ (0) fu (zξ1 |s ) = min fs (zξ1 , r|s(0) ), 1 gs (r) dr.
(56)
0
From (9), (47) and (52), we have
fs (zξ1 , r|s(0) ) =
X
Bδd
δd1 ,δd2 ,δd
=
X δd
=
1
,δd2 ,δd |s(0) p2 (zξ1 , r, δd1 , δd2 , δd )
Aδd |s(0) p2 (zξ1 , r,
X
√
Ad p2 (zξ1 , r, d),
n,
√
n, δd ) (57)
1≤d≤n
May 11, 2014
DRAFT
20
where
p2 (zξ1 , r, d) =
√
1−
and
Γ( n−1 ) 2 π Γ( n−2 ) 2
Γ( n−1 ) 2 √ π Γ( n−2 ) 2
R arccos( βd (zξ1 ) ) r
0
R
|βd (zξ )| 1 ) arccos( r
0
fu (zξ1 ) =
X s
√
√
n
√
n √ , (58) n−3 sin φ dφ, r ≥ |βd (zξ1 )|, zξ1 ≥ n √ 1, r < |βd (zξ1 )|, zξ1 ≥ n 0, r < βd (zξ1 ), zξ1
